Thus, by the monotonicity of expectations (see for example [1]}), one has
\(\mathop {\sum }\nolimits _{i=1}^\infty p_i \mathop {\rm liminf}_{\begin{array}{c}\nu \rightarrow \infty \\ |j-i|\rightarrow 0\end{array}} g^\nu (j) \ge \mathop {\sum }\nolimits _{i=1}^\infty p_i g(i).\) 
